On the value distribution of a differential monomial and some normality criteria

The aim of this paper is to study the zero distribution of the differential polynomial $\displaystyle af^{q_{0}}(f')^{q_{1}}...(f^{(k)})^{q_{k}}-\varphi,$where $f$ is a transcendental meromorphic function and $a=a(z)(\not\equiv 0,\infty)$ and $\varphi(\not\equiv 0,\infty)$ are small functions of $f$. Moreover, using this value distribution result, we prove the following normality criterion for family of analytic functions:\\ {\it Let $\mathscr{F}$ be a family of analytic functions on a domain $D$ and let $k \geq1$, $q_{0}\geq 2$, $q_{i} \geq 0$ $(i=1,2,\ldots,k-1)$, $q_{k}\geq 1$ be positive integers. If for each $f\in \mathscr{F}$: i.\ $f$ has only zeros of multiplicity at least $k$,\ ii.\ $\displaystyle f^{q_{0}}(f')^{q_{1}}\ldots(f^{(k)})^{q_{k}}\not=1$,then $\mathscr{F}$ is normal on domain $D$.

Entire functions of restricted hyper-order sharing a set of two small functions IM with their linear c-shift operators

PurposeThe paper aims to build the relationship between an entire function of restricted hyper-order with its linear c-shift operator.Design/methodology/approachStandard methodology for papers in difference and shift operators and value distribution theory have been used.FindingsThe relation between an entire function of restricted hyper-order with its linear c-shift operator was found under the periphery of sharing a set of two small functions IM (ignoring multiplicities) when exponent of convergence of zeros is strictly less than its order. This research work is an improvement and extension of two previous papers.Originality/valueThis is an original research work.

ON BOREL DIRECTION CONCERNING SMALL FUNCTIONS

In this paper, we shall prove Theorem 1 Let $f$ be nonconstant meromorphic in $\mathbb{C}$ with finite positive order $\lambda$, $\lambda(r)$ be a proximate order of $f$ and $U(r, f)=r^{\lambda(r)}$, then for each number $\alpha$,$0<\alpha<\pi/2$, there exists a number $\phi_0$ with $0\le \phi_0 < 2\pi$ such that the inequality \[ \limsup_{r\to\infty}\sum_{i=1}^3 n(r, \phi_0, \alpha, f=a_i(z))/U(r, f)>0,\] holds for any three distinct meromorphic function $a_i(z)(i=1, 2, 3)$ with $T(r,a_i)=o(U(r, f))$ as $r\to\infty$.

Two meromorphic functions on annuli sharing some pairs of small functions or values

<abstract><p>In this paper, we prove that two admissible meromorphic functions on an annulus must be linked by a quasi-Möbius transformation if they share some pairs of small function with multiplicities truncated by $ 4 $. We also give the representation of Möbius transformation between two admissible meromorphic functions on an annulus if they share four pairs of values with multiplicities truncated by $ 4 $. In our results, the zeros with multiplicities more than a certain number are not needed to be counted if their multiplicities are bigger than a certain number.</p></abstract>

Entire functions that share two pairs of small functions

In this paper, we study the unicity of entire functions and their derivatives and obtain the following result: let f f be a non-constant entire function, let a 1 {a}_{1} , a 2 {a}_{2} , b 1 {b}_{1} , and b 2 {b}_{2} be four small functions of f f such that a 1 ≢ b 1 {a}_{1}\not\equiv {b}_{1} , a 2 ≢ b 2 {a}_{2}\not\equiv {b}_{2} , and none of them is identically equal to ∞ \infty . If f f and f ( k ) {f}^{\left(k)} share ( a 1 , a 2 ) \left({a}_{1},{a}_{2}) CM and share ( b 1 , b 2 ) \left({b}_{1},{b}_{2}) IM, then ( a 2 − b 2 ) f − ( a 1 − b 1 ) f ( k ) ≡ a 2 b 1 − a 1 b 2 \left({a}_{2}-{b}_{2})f-\left({a}_{1}-{b}_{1}){f}^{\left(k)}\equiv {a}_{2}{b}_{1}-{a}_{1}{b}_{2} . This extends the result due to Li and Yang [Value sharing of an entire function and its derivatives, J. Math. Soc. Japan. 51 (1999), no. 7, 781–799].